Accurate trajectory prediction is crucial for safe and efficient autonomous driving, but handling partial observations presents significant challenges. To address this, we propose a novel trajectory prediction framework called Partial Observations Prediction (POP) for congested urban road scenarios. The framework consists of two key stages: self-supervised learning (SSL) and feature distillation. POP first employs SLL to help the model learn to reconstruct history representations, and then utilizes feature distillation as the fine-tuning task to transfer knowledge from the teacher model, which has been pre-trained with complete observations, to the student model, which has only few observations. POP achieves comparable results to top-performing methods in open-loop experiments and outperforms the baseline method in closed-loop simulations, including safety metrics. Qualitative results illustrate the superiority of POP in providing reasonable and safe trajectory predictions.